Eisenstein-Kronecker series via the Poincar\'e bundle

TL;DR

This paper offers a new algebraic approach to Eisenstein--Kronecker series and introduces a novel construction of Katz's $p$-adic Eisenstein measure using the Poincaré bundle and $p$-adic theta functions.

Contribution

It provides an alternative algebraic construction of Eisenstein--Kronecker series and a new conceptual framework for Katz's $p$-adic Eisenstein measure.

Findings

New algebraic construction of Eisenstein--Kronecker series via Poincaré bundle

A novel $p$-adic Eisenstein measure using $p$-adic theta functions

Enhanced understanding of algebraic and $p$-adic properties of Eisenstein series

Abstract

A classical construction of Katz gives a purely algebraic construction of Eisenstein--Kronecker series using the Gau\ss--Manin connection on the universal elliptic curve. This approach gives a systematic way to study algebraic and $p$-adic properties of real-analytic Eisenstein series. In the first part of this paper we provide an alternative algebraic construction of Eisenstein--Kronecker series via the Poincar\'e bundle. Building on this, we give in the second part a new conceptional construction of Katz' two-variable $p$-adic Eisenstein measure through $p$-adic theta functions of the Poincar\'e bundle.